Theorems · Theorem · functional analysis
IsSelfAdjoint.toReal_spectralRadius_complex_eq_norm
∀ {A : Type u_1} [inst : CStarAlgebra A] {a : A}, IsSelfAdjoint a → (spectralRadius ℂ a).toReal = ‖a‖In a C⋆-algebra, the spectral radius of a self-adjoint element is equal to its norm.
See IsSelfAdjoint.toReal_spectralRadius_eq_norm for a version involving
spectralRadius ℝ a.
- Defined in
- Mathlib.Analysis.CStarAlgebra.Spectrum
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 288 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CStarAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Complexstatement · cited by 5,565
- Norm.normstatement and proof · cited by 5,413
- ENNReal.toRealstatement and proof · cited by 859
- IsSelfAdjointstatement and proof · cited by 545
- CStarAlgebrastatement and proof · cited by 123
- spectralRadiusstatement · cited by 36
- IsSelfAdjoint.spectralRadius_eq_nnnormproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- CStarAlgebra.toReal_spectralRadius_star_mul_self_eq_norm_sqproof · cited by 2